Related papers: Bichromatic lines in the plane

Norine's antipodal-colouring conjecture, in a form given by Feder and Subi, asserts that whenever the edges of the discrete cube are 2-coloured there must exist a path between two opposite vertices along which there is at most one colour…

We show that any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic sets of the form $\{x,y,xy,x+y\},$ and more generally monochromatic sets of the form $\{x_i,\prod x_i,\sum x_i: i\leq k\}$ for any $k\in\mathbb{N}.$ Along…

Answering a question raised by Dudek and Pra\l{}at, we show that if $pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2/3+o(1))$. This result is optimal in the sense that…

We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same…

A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to…

We study the infinite graph of $n$-dimensional rectangular grid that doesn't appear distance regular and the distance regular colorings of this graph, which are defined as the distance colorings with respect to completely regular codes. It…

The Cyclic Coloring Conjecture asserts that the vertices of every plane graph with maximum face size D can be colored using at most 3D/2 colors in such a way that no face is incident with two vertices of the same color. The Cyclic Coloring…

A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors…

Consider a 2D coordinate space with a set of red and a set of blue points. We define a scenic point as a point that is equidistant to a red point and a blue point. The set of contiguous scenic points form a scenic path. The perpendicular…

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of $G$ is the maximum integer $b(G)$ for which $G$ has a b-coloring…

Given D and H two digraphs, D is H-coloured iff the arcs of D are coloured with the vertices of H. After defining what do we mean by an H-walk in the coloured D, we characterise those H, which we call panchromatic patterns, for which all D…

By the Road Coloring Theorem (Trahtman, 2008), the edges of any aperiodic directed multigraph with a constant out-degree can be colored such that the resulting automaton admits a reset word. There may also be a need for a particular reset…

A clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colours in such a colouring is the clique chromatic number. In this paper, we study…

Kelly's theorem states that a set of $n$ points affinely spanning $\mathbb{C}^3$ must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least…

Let n>0 be a number. Let Gn be the graph on n-dimensional Euclidean space connecting points of rational distance. It is consistent with the choiceless theory ZF+DC that Gn has countable chromatic number yet Gn+1 does not.

In this paper, we introduce the notion of 2-boundary planar graphs. A graph is 2-boundary planar if it has an embedding in the plane so that all vertices lie on the boundary of at most two faces and no edges are crossed. A proper coloring…

Maximal planar graph refers to the planar graph with the most edges, which means no more edges can be added so that the resulting graph is still planar. The Four-Color Conjecture says that every planar graph without loops is 4-colorable.…

We study the number of monochromatic solution to linear equation in $\{1,\dots,n\}$ when we color the set by at least three colors. We consider the $r$-commonness for $r\geq 3$ of linear equation with odd number of terms, and we also prove…

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph $G$ is the largest integer $k$ such that $G$ admits a…

Let $P$ be a set of $n$ points in strictly convex position in the plane. Let $D_n$ be the graph whose vertex set is the set of all line segments with endpoints in $P$, where disjoint segments are adjacent. The chromatic number of this graph…